The answer to question 1 is actually positive, accomplished by Feferman set theory. I have learned of (a variant of) it from Cantor's Attic.

We define the theory FST by extending ZFC by a new predicate symbol $C$ and the following axioms concerning it:

• $C$ defines a closed and unbounded class of cardinals,
• For any formula $\varphi$ with one parameter, we have an axiom $$\forall\kappa\in C\forall x\in V_\kappa\big(\phi(x)\iff (V_\kappa\models\phi(x))\big).$$

That is, all elements of $C$ are correct cardinals, so that $V_\kappa$ is an elementary substructure of $V$.

The reason this extension is conservative is the reflection theorem, which asserts that for any finite collection $\varphi=(\varphi_1,\dots,\varphi_n)$ of predicates, there is a closed unbounded class of cardinals $C_\varphi$ satisfying the axioms above for all $\varphi_i$. Since any proof in FST can only invoke finitely many instances of this axiom schema, we can repeat this proof in ZFC using the intersection of those classes $C_\varphi$ in place of $C$.

It is not hard to see that cardinals in $C$ are strong limit cardinals. However, they may fail to be regular - the axiom schema merely guarantees that any $\kappa\in C$ is regular "from within $V_\kappa$", that is, any subset of $\kappa$ which is definable (as a class) in $V_\kappa$ must have length $\kappa$. This shortcoming is I imagine why this theory has not gained much traction, since although you are unlikely to run into such undefinable classes inside $V_\kappa$, in principle it requires some bookkeeping to make sure things do not go wrong. See this MO post for a similar discussion.

On the other hand, correctness of these cardinals has its advantages - we can work with all of $V$ or internally to some $V_\kappa,\kappa\in C$, and the results we prove will be equally valid by the above axiom schema.

The answer to question 1 is actually positive, accomplished by Feferman set theory. I have learned of (a variant of) it from Cantor's Attic.

We define the theory FST by taking ZFC, adjoining a new predicate symbol $C$ to its language, and adding the following axioms concerning it:

• $C$ defines a closed and unbounded class of cardinals,
• For any formula $\varphi$ in the language of ZFC (i.e. not involving $C$) with one parameter, we have an axiom $$\forall\kappa\in C\forall x\in V_\kappa\big(\phi(x)\iff (V_\kappa\models\phi(x))\big).$$

That is, any element $\kappa\in C$ is a correct cardinal, so that $V_\kappa$ is an elementary substructure of $V$.

The reason this extension is conservative is the reflection theorem, which asserts that for any finite collection $\varphi=(\varphi_1,\dots,\varphi_n)$ of predicates, there is a closed unbounded class of cardinals $C_\varphi$ satisfying the axioms above for all $\varphi_i$. Since any proof in FST can only invoke finitely many instances of this axiom schema, we can repeat this proof in ZFC using the class $C_\varphi$ in place of $C$.

It is not hard to see that cardinals in $C$ are strong limit cardinals. However, they may fail to be regular - the axiom schema merely guarantees that any $\kappa\in C$ is regular "from within $V_\kappa$", that is, any subset of $\kappa$ which is definable in $V_\kappa$ (as a class) must have length $\kappa$. This shortcoming is I imagine why this theory has not gained much traction, since although you are unlikely to run into such undefinable classes inside $V_\kappa$, in principle it requires some bookkeeping to make sure things do not go wrong. See this MO post for a similar discussion.

On the other hand, correctness of these cardinals has its advantages - we can work with all of $V$ or internally to some $V_\kappa,\kappa\in C$, and the results we prove will be equally valid by the above axiom schema.